Since your space has codimension $1$ (i.e., dimension $3$ in a $4$-dimensional space), the following recipe will work: find a vector $\tilde e_4$ orthogonal to your subspace (try $(6,-2,4,-10)$), extend to an orthogonal basis $(\tilde e_i)_{i=1,\ldots,4}$ an normalise $e_i=\tilde e_i/ |\tilde e_i|$. Now the coordinates of a vector on this basis are given by scalar products: $v=\sum_{i=1}^4\langle e_i,v\rangle e_i$, and if you leave out $e_4$ you get the orthogonal projection $\sum_{i=1}^3\langle e_i,v\rangle e_i$ of $v$ onto your subspace. In matrix form, take the matrix with columns the coordinates of the $e_i$, and multiply by the matrix with as first three rows the coordinates of $e_1,e_2,e_3$ and last row zero.

Additional exercise: try to find out how to avoid introducing the ugly square roots of the normalization (which should disappear in the final matrix, since your projection is defined over $\Bbb Q$).